The different parameterizations that were considered were the Boyd, Grinstein, and Lebed (BGL) parameterization [1]}, the Bourrely, Caprini, and Lellouch (BCL) parameterization [2]}, and the Caprini, Lellouch, and Neubert (CLN) parameterization [3]}. The BGL and BCL parameterizations both incorporate the z-expansion, while the CLN does not. The form factors, and ultimately the differential decay rate, are functions of the lepton momentum transfer variable \(q^2\) . A problem with this is that there is a branch cut in the complex \(q^2\)  plane. To get form factors that are analytic in the first sheet, a new variable is needed. The z-expansion maps the entire \(q^2\)  plane onto the unit disk in the complex-\(z\)  plane, with the branch cut being mapped onto the boundary of the disk [4]}. This allows the form factors to be written as a simple power series of \(z\) .
